用线性方法解非线性系统动力学，提升控制精度与效率
Koopman operator theory: fundamentals, control, and applications
- 通过观测量函数将非线性系统转化为线性表示
- 基于数据的有限维近似具有可验证的误差界
- 适合控制系统设计，尤其适用于复杂动态建模
Koopman算子因其能对高度复杂的动力系统提供全局线性表示而受到广泛关注。该算子通过实值或复值可观测函数，以线性方式描述非线性动态。近年来提出的数据驱动方法，如扩展动态模态分解（EDMD）、其核化变体及机器学习技术，可生成有限维近似，并附带有限数据下的误差界。本文是一篇教程性质的综述，简明介绍Koopman算子理论及其在系统与控制中的应用。重点聚焦于数据驱动的代理模型、含输入系统的扩展，以及基于Koopman理论的控制器设计。此外，演示了关键方法：EDMD与Koopman MPC。为此，我们提供了包含源代码的仿真研究，使读者可逐步体验该理论在系统与控制中的实际应用。
原文摘要 · Abstract (English)
The Koopman operator has gained considerable attention due to its ability to provide a global linear representation of highly complex dynamical systems. The operator describes nonlinear dynamics in a linear way through the lens of real- or complex-valued observable functions. Recently proposed data-driven techniques, like extended dynamic mode decomposition (EDMD), its kernelized variant, and machine-learning methods, can be used to generate finite-dimensional approximations accompanied by finite-data error bounds. In this tutorial paper, we provide a concise introduction into Koopman operator theory and its use in systems and control. A particular focus is put on data-driven surrogate models, their extension to systems with inputs, and controller design using Koopman operator theory. Moreover, we demonstrate the key techniques, i.e., EDMD and Koopman MPC. To this end, we provide simulation studies including source code on GitHub to enable the interested reader to experience the Koopman operator in systems and control step by step.
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